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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    <a name="line.17"></a>
<FONT color="green">018</FONT>    package org.apache.commons.math3.linear;<a name="line.18"></a>
<FONT color="green">019</FONT>    <a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math3.util.FastMath;<a name="line.20"></a>
<FONT color="green">021</FONT>    <a name="line.21"></a>
<FONT color="green">022</FONT>    /**<a name="line.22"></a>
<FONT color="green">023</FONT>     * Calculates the rectangular Cholesky decomposition of a matrix.<a name="line.23"></a>
<FONT color="green">024</FONT>     * &lt;p&gt;The rectangular Cholesky decomposition of a real symmetric positive<a name="line.24"></a>
<FONT color="green">025</FONT>     * semidefinite matrix A consists of a rectangular matrix B with the same<a name="line.25"></a>
<FONT color="green">026</FONT>     * number of rows such that: A is almost equal to BB&lt;sup&gt;T&lt;/sup&gt;, depending<a name="line.26"></a>
<FONT color="green">027</FONT>     * on a user-defined tolerance. In a sense, this is the square root of A.&lt;/p&gt;<a name="line.27"></a>
<FONT color="green">028</FONT>     * &lt;p&gt;The difference with respect to the regular {@link CholeskyDecomposition}<a name="line.28"></a>
<FONT color="green">029</FONT>     * is that rows/columns may be permuted (hence the rectangular shape instead<a name="line.29"></a>
<FONT color="green">030</FONT>     * of the traditional triangular shape) and there is a threshold to ignore<a name="line.30"></a>
<FONT color="green">031</FONT>     * small diagonal elements. This is used for example to generate {@link<a name="line.31"></a>
<FONT color="green">032</FONT>     * org.apache.commons.math3.random.CorrelatedRandomVectorGenerator correlated<a name="line.32"></a>
<FONT color="green">033</FONT>     * random n-dimensions vectors} in a p-dimension subspace (p &lt; n).<a name="line.33"></a>
<FONT color="green">034</FONT>     * In other words, it allows generating random vectors from a covariance<a name="line.34"></a>
<FONT color="green">035</FONT>     * matrix that is only positive semidefinite, and not positive definite.&lt;/p&gt;<a name="line.35"></a>
<FONT color="green">036</FONT>     * &lt;p&gt;Rectangular Cholesky decomposition is &lt;em&gt;not&lt;/em&gt; suited for solving<a name="line.36"></a>
<FONT color="green">037</FONT>     * linear systems, so it does not provide any {@link DecompositionSolver<a name="line.37"></a>
<FONT color="green">038</FONT>     * decomposition solver}.&lt;/p&gt;<a name="line.38"></a>
<FONT color="green">039</FONT>     *<a name="line.39"></a>
<FONT color="green">040</FONT>     * @see &lt;a href="http://mathworld.wolfram.com/CholeskyDecomposition.html"&gt;MathWorld&lt;/a&gt;<a name="line.40"></a>
<FONT color="green">041</FONT>     * @see &lt;a href="http://en.wikipedia.org/wiki/Cholesky_decomposition"&gt;Wikipedia&lt;/a&gt;<a name="line.41"></a>
<FONT color="green">042</FONT>     * @version $Id: RectangularCholeskyDecomposition.java 1422313 2012-12-15 18:53:41Z psteitz $<a name="line.42"></a>
<FONT color="green">043</FONT>     * @since 2.0 (changed to concrete class in 3.0)<a name="line.43"></a>
<FONT color="green">044</FONT>     */<a name="line.44"></a>
<FONT color="green">045</FONT>    public class RectangularCholeskyDecomposition {<a name="line.45"></a>
<FONT color="green">046</FONT>    <a name="line.46"></a>
<FONT color="green">047</FONT>        /** Permutated Cholesky root of the symmetric positive semidefinite matrix. */<a name="line.47"></a>
<FONT color="green">048</FONT>        private final RealMatrix root;<a name="line.48"></a>
<FONT color="green">049</FONT>    <a name="line.49"></a>
<FONT color="green">050</FONT>        /** Rank of the symmetric positive semidefinite matrix. */<a name="line.50"></a>
<FONT color="green">051</FONT>        private int rank;<a name="line.51"></a>
<FONT color="green">052</FONT>    <a name="line.52"></a>
<FONT color="green">053</FONT>        /**<a name="line.53"></a>
<FONT color="green">054</FONT>         * Decompose a symmetric positive semidefinite matrix.<a name="line.54"></a>
<FONT color="green">055</FONT>         * &lt;p&gt;<a name="line.55"></a>
<FONT color="green">056</FONT>         * &lt;b&gt;Note:&lt;/b&gt; this constructor follows the linpack method to detect dependent<a name="line.56"></a>
<FONT color="green">057</FONT>         * columns by proceeding with the Cholesky algorithm until a nonpositive diagonal<a name="line.57"></a>
<FONT color="green">058</FONT>         * element is encountered.<a name="line.58"></a>
<FONT color="green">059</FONT>         *<a name="line.59"></a>
<FONT color="green">060</FONT>         * @see &lt;a href="http://eprints.ma.man.ac.uk/1193/01/covered/MIMS_ep2008_56.pdf"&gt;<a name="line.60"></a>
<FONT color="green">061</FONT>         * Analysis of the Cholesky Decomposition of a Semi-definite Matrix&lt;/a&gt;<a name="line.61"></a>
<FONT color="green">062</FONT>         *<a name="line.62"></a>
<FONT color="green">063</FONT>         * @param matrix Symmetric positive semidefinite matrix.<a name="line.63"></a>
<FONT color="green">064</FONT>         * @exception NonPositiveDefiniteMatrixException if the matrix is not<a name="line.64"></a>
<FONT color="green">065</FONT>         * positive semidefinite.<a name="line.65"></a>
<FONT color="green">066</FONT>         * @since 3.1<a name="line.66"></a>
<FONT color="green">067</FONT>         */<a name="line.67"></a>
<FONT color="green">068</FONT>        public RectangularCholeskyDecomposition(RealMatrix matrix)<a name="line.68"></a>
<FONT color="green">069</FONT>            throws NonPositiveDefiniteMatrixException {<a name="line.69"></a>
<FONT color="green">070</FONT>            this(matrix, 0);<a name="line.70"></a>
<FONT color="green">071</FONT>        }<a name="line.71"></a>
<FONT color="green">072</FONT>    <a name="line.72"></a>
<FONT color="green">073</FONT>        /**<a name="line.73"></a>
<FONT color="green">074</FONT>         * Decompose a symmetric positive semidefinite matrix.<a name="line.74"></a>
<FONT color="green">075</FONT>         *<a name="line.75"></a>
<FONT color="green">076</FONT>         * @param matrix Symmetric positive semidefinite matrix.<a name="line.76"></a>
<FONT color="green">077</FONT>         * @param small Diagonal elements threshold under which columns are<a name="line.77"></a>
<FONT color="green">078</FONT>         * considered to be dependent on previous ones and are discarded.<a name="line.78"></a>
<FONT color="green">079</FONT>         * @exception NonPositiveDefiniteMatrixException if the matrix is not<a name="line.79"></a>
<FONT color="green">080</FONT>         * positive semidefinite.<a name="line.80"></a>
<FONT color="green">081</FONT>         */<a name="line.81"></a>
<FONT color="green">082</FONT>        public RectangularCholeskyDecomposition(RealMatrix matrix, double small)<a name="line.82"></a>
<FONT color="green">083</FONT>            throws NonPositiveDefiniteMatrixException {<a name="line.83"></a>
<FONT color="green">084</FONT>    <a name="line.84"></a>
<FONT color="green">085</FONT>            final int order = matrix.getRowDimension();<a name="line.85"></a>
<FONT color="green">086</FONT>            final double[][] c = matrix.getData();<a name="line.86"></a>
<FONT color="green">087</FONT>            final double[][] b = new double[order][order];<a name="line.87"></a>
<FONT color="green">088</FONT>    <a name="line.88"></a>
<FONT color="green">089</FONT>            int[] index = new int[order];<a name="line.89"></a>
<FONT color="green">090</FONT>            for (int i = 0; i &lt; order; ++i) {<a name="line.90"></a>
<FONT color="green">091</FONT>                index[i] = i;<a name="line.91"></a>
<FONT color="green">092</FONT>            }<a name="line.92"></a>
<FONT color="green">093</FONT>    <a name="line.93"></a>
<FONT color="green">094</FONT>            int r = 0;<a name="line.94"></a>
<FONT color="green">095</FONT>            for (boolean loop = true; loop;) {<a name="line.95"></a>
<FONT color="green">096</FONT>    <a name="line.96"></a>
<FONT color="green">097</FONT>                // find maximal diagonal element<a name="line.97"></a>
<FONT color="green">098</FONT>                int swapR = r;<a name="line.98"></a>
<FONT color="green">099</FONT>                for (int i = r + 1; i &lt; order; ++i) {<a name="line.99"></a>
<FONT color="green">100</FONT>                    int ii  = index[i];<a name="line.100"></a>
<FONT color="green">101</FONT>                    int isr = index[swapR];<a name="line.101"></a>
<FONT color="green">102</FONT>                    if (c[ii][ii] &gt; c[isr][isr]) {<a name="line.102"></a>
<FONT color="green">103</FONT>                        swapR = i;<a name="line.103"></a>
<FONT color="green">104</FONT>                    }<a name="line.104"></a>
<FONT color="green">105</FONT>                }<a name="line.105"></a>
<FONT color="green">106</FONT>    <a name="line.106"></a>
<FONT color="green">107</FONT>    <a name="line.107"></a>
<FONT color="green">108</FONT>                // swap elements<a name="line.108"></a>
<FONT color="green">109</FONT>                if (swapR != r) {<a name="line.109"></a>
<FONT color="green">110</FONT>                    final int tmpIndex    = index[r];<a name="line.110"></a>
<FONT color="green">111</FONT>                    index[r]              = index[swapR];<a name="line.111"></a>
<FONT color="green">112</FONT>                    index[swapR]          = tmpIndex;<a name="line.112"></a>
<FONT color="green">113</FONT>                    final double[] tmpRow = b[r];<a name="line.113"></a>
<FONT color="green">114</FONT>                    b[r]                  = b[swapR];<a name="line.114"></a>
<FONT color="green">115</FONT>                    b[swapR]              = tmpRow;<a name="line.115"></a>
<FONT color="green">116</FONT>                }<a name="line.116"></a>
<FONT color="green">117</FONT>    <a name="line.117"></a>
<FONT color="green">118</FONT>                // check diagonal element<a name="line.118"></a>
<FONT color="green">119</FONT>                int ir = index[r];<a name="line.119"></a>
<FONT color="green">120</FONT>                if (c[ir][ir] &lt;= small) {<a name="line.120"></a>
<FONT color="green">121</FONT>    <a name="line.121"></a>
<FONT color="green">122</FONT>                    if (r == 0) {<a name="line.122"></a>
<FONT color="green">123</FONT>                        throw new NonPositiveDefiniteMatrixException(c[ir][ir], ir, small);<a name="line.123"></a>
<FONT color="green">124</FONT>                    }<a name="line.124"></a>
<FONT color="green">125</FONT>    <a name="line.125"></a>
<FONT color="green">126</FONT>                    // check remaining diagonal elements<a name="line.126"></a>
<FONT color="green">127</FONT>                    for (int i = r; i &lt; order; ++i) {<a name="line.127"></a>
<FONT color="green">128</FONT>                        if (c[index[i]][index[i]] &lt; -small) {<a name="line.128"></a>
<FONT color="green">129</FONT>                            // there is at least one sufficiently negative diagonal element,<a name="line.129"></a>
<FONT color="green">130</FONT>                            // the symmetric positive semidefinite matrix is wrong<a name="line.130"></a>
<FONT color="green">131</FONT>                            throw new NonPositiveDefiniteMatrixException(c[index[i]][index[i]], i, small);<a name="line.131"></a>
<FONT color="green">132</FONT>                        }<a name="line.132"></a>
<FONT color="green">133</FONT>                    }<a name="line.133"></a>
<FONT color="green">134</FONT>    <a name="line.134"></a>
<FONT color="green">135</FONT>                    // all remaining diagonal elements are close to zero, we consider we have<a name="line.135"></a>
<FONT color="green">136</FONT>                    // found the rank of the symmetric positive semidefinite matrix<a name="line.136"></a>
<FONT color="green">137</FONT>                    loop = false;<a name="line.137"></a>
<FONT color="green">138</FONT>    <a name="line.138"></a>
<FONT color="green">139</FONT>                } else {<a name="line.139"></a>
<FONT color="green">140</FONT>    <a name="line.140"></a>
<FONT color="green">141</FONT>                    // transform the matrix<a name="line.141"></a>
<FONT color="green">142</FONT>                    final double sqrt = FastMath.sqrt(c[ir][ir]);<a name="line.142"></a>
<FONT color="green">143</FONT>                    b[r][r] = sqrt;<a name="line.143"></a>
<FONT color="green">144</FONT>                    final double inverse  = 1 / sqrt;<a name="line.144"></a>
<FONT color="green">145</FONT>                    final double inverse2 = 1 / c[ir][ir];<a name="line.145"></a>
<FONT color="green">146</FONT>                    for (int i = r + 1; i &lt; order; ++i) {<a name="line.146"></a>
<FONT color="green">147</FONT>                        final int ii = index[i];<a name="line.147"></a>
<FONT color="green">148</FONT>                        final double e = inverse * c[ii][ir];<a name="line.148"></a>
<FONT color="green">149</FONT>                        b[i][r] = e;<a name="line.149"></a>
<FONT color="green">150</FONT>                        c[ii][ii] -= c[ii][ir] * c[ii][ir] * inverse2;<a name="line.150"></a>
<FONT color="green">151</FONT>                        for (int j = r + 1; j &lt; i; ++j) {<a name="line.151"></a>
<FONT color="green">152</FONT>                            final int ij = index[j];<a name="line.152"></a>
<FONT color="green">153</FONT>                            final double f = c[ii][ij] - e * b[j][r];<a name="line.153"></a>
<FONT color="green">154</FONT>                            c[ii][ij] = f;<a name="line.154"></a>
<FONT color="green">155</FONT>                            c[ij][ii] = f;<a name="line.155"></a>
<FONT color="green">156</FONT>                        }<a name="line.156"></a>
<FONT color="green">157</FONT>                    }<a name="line.157"></a>
<FONT color="green">158</FONT>    <a name="line.158"></a>
<FONT color="green">159</FONT>                    // prepare next iteration<a name="line.159"></a>
<FONT color="green">160</FONT>                    loop = ++r &lt; order;<a name="line.160"></a>
<FONT color="green">161</FONT>                }<a name="line.161"></a>
<FONT color="green">162</FONT>            }<a name="line.162"></a>
<FONT color="green">163</FONT>    <a name="line.163"></a>
<FONT color="green">164</FONT>            // build the root matrix<a name="line.164"></a>
<FONT color="green">165</FONT>            rank = r;<a name="line.165"></a>
<FONT color="green">166</FONT>            root = MatrixUtils.createRealMatrix(order, r);<a name="line.166"></a>
<FONT color="green">167</FONT>            for (int i = 0; i &lt; order; ++i) {<a name="line.167"></a>
<FONT color="green">168</FONT>                for (int j = 0; j &lt; r; ++j) {<a name="line.168"></a>
<FONT color="green">169</FONT>                    root.setEntry(index[i], j, b[i][j]);<a name="line.169"></a>
<FONT color="green">170</FONT>                }<a name="line.170"></a>
<FONT color="green">171</FONT>            }<a name="line.171"></a>
<FONT color="green">172</FONT>    <a name="line.172"></a>
<FONT color="green">173</FONT>        }<a name="line.173"></a>
<FONT color="green">174</FONT>    <a name="line.174"></a>
<FONT color="green">175</FONT>        /** Get the root of the covariance matrix.<a name="line.175"></a>
<FONT color="green">176</FONT>         * The root is the rectangular matrix &lt;code&gt;B&lt;/code&gt; such that<a name="line.176"></a>
<FONT color="green">177</FONT>         * the covariance matrix is equal to &lt;code&gt;B.B&lt;sup&gt;T&lt;/sup&gt;&lt;/code&gt;<a name="line.177"></a>
<FONT color="green">178</FONT>         * @return root of the square matrix<a name="line.178"></a>
<FONT color="green">179</FONT>         * @see #getRank()<a name="line.179"></a>
<FONT color="green">180</FONT>         */<a name="line.180"></a>
<FONT color="green">181</FONT>        public RealMatrix getRootMatrix() {<a name="line.181"></a>
<FONT color="green">182</FONT>            return root;<a name="line.182"></a>
<FONT color="green">183</FONT>        }<a name="line.183"></a>
<FONT color="green">184</FONT>    <a name="line.184"></a>
<FONT color="green">185</FONT>        /** Get the rank of the symmetric positive semidefinite matrix.<a name="line.185"></a>
<FONT color="green">186</FONT>         * The r is the number of independent rows in the symmetric positive semidefinite<a name="line.186"></a>
<FONT color="green">187</FONT>         * matrix, it is also the number of columns of the rectangular<a name="line.187"></a>
<FONT color="green">188</FONT>         * matrix of the decomposition.<a name="line.188"></a>
<FONT color="green">189</FONT>         * @return r of the square matrix.<a name="line.189"></a>
<FONT color="green">190</FONT>         * @see #getRootMatrix()<a name="line.190"></a>
<FONT color="green">191</FONT>         */<a name="line.191"></a>
<FONT color="green">192</FONT>        public int getRank() {<a name="line.192"></a>
<FONT color="green">193</FONT>            return rank;<a name="line.193"></a>
<FONT color="green">194</FONT>        }<a name="line.194"></a>
<FONT color="green">195</FONT>    <a name="line.195"></a>
<FONT color="green">196</FONT>    }<a name="line.196"></a>




























































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